Asymptotically, one reaches the vacuum when the asymptotic spacetime is either AdS\(_3\)  or Minkowski; these are maximally symmetric spacetimes that can be viewed as the ground states of General Relativity [1]}. In this asymptotic region, the kinetic term for the scalar field and gauge field tend to zero: \(-\frac{1}{2}(D_{\mu } \phi )^{\dagger }(D^{\mu } \phi )\rightarrow 0\)  and \(-\frac{1}{4} F_{\mu \nu }F^{\mu \nu }\rightarrow 0\) . This occurs when asymptotically the magnitude of the scalar field approaches the minimum of the potential (the non-zero VEV) and the gauge field approaches a non-zero constant equal to the winding number \(n\) . In \(2+1\)  dimensions, the asymptotic value of the Ricci scalar is given byNote that the vacuum Einstein field equations with cosmological constant \(\Lambda _{eff}\)  yield \(R=4 \Lambda _{eff}\)  in \(3+1\)  dimensions but \(R=6 \Lambda _{eff}\)  in \(2+1\)  dimensions. \(6\, \Lambda _{eff}\,\)  where \(\Lambda _{eff}\)  is either negative (AdS\(_3\)  background) or zero (Minkowski background). The potential for the scalar field can be readily picked out from the Lagrangian and asymptotically is given by
\(V(|\phi |)=\dfrac{\lambda }{4}(|\phi |^2-v^2)^2-\xi \, R\,|\phi |^2=\dfrac{\lambda }{4}(|\phi |^2-v^2)^2 - 6\,\xi \,\Lambda _{eff} |\phi |^2\,.\) 
